Observe that by the properties (i) and (iii) above, the labelled tree associated with a bipartite planar map having \(n-1\)  edges and \(k+1\)  vertices with one distinguished vertex has \(n\)  vertices and \(k\)  leaves.
The last property shows that controlling the labels on the tree is equivalent to controlling distances to the distinguished vertex in the map. It then remains to control all the pairwise distances, which is particularly challenging and was done for quadrangulations, by Le Gall [1]} and Miermont [2]}. Thanks to their work, going from the control of labels on the tree to that of all distances in the map, in the case of the Brownian map, is now rather simple.
We will in fact not consider the labels here, but only the tree itself, and simply rely on general results from [3]}.
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